GRÖBNER BASES AND DETERMINANTAL IDEALS -- An Introduction

We give an introduction to the theory of determinantal ideals and rings, their Gröbner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the Knuth-Robinson-Schensted correspondence. The article contains a section treating the basic results about the passage to initial ideals and algebras. Let K be a field and X an m × n matrix of indeterminates over K. For a given positive integer t ≤ min(m, n), we consider the ideal I t = I t (X) generated by the t-minors (i. e. the determinants of the t × t submatrices) of X in the polynomial ring K[X ] generated by all the indeterminates X i j. From the viewpoint of algebraic geometry K[X ] should be regarded as the coordinate ring of the variety of K-linear maps f : K m → K n. Then V (I t) is just the variety of all f such that rank f < t, and K[X ]/I t is its coordinate ring. The study of the determinantal ideals I t and the objects related to them has numerous connections with invariant theory, representation theory, and combina-torics. For a detailed account we refer the reader to Bruns and Vetter [17]. A large part of the theory of determinantal ideals can be developed over the ring Z of integers (instead of a base field K) and then transferred to arbitrary rings B of coefficients (see [17]). For simplicity we restrict ourselves to fields. This article follows the line of investigation started by Sturmfels' article [65] in which he applied the Knuth-Robinson-Schensted correspondence KRS (Knuth [50]) to the study of the determinantal ideals I t. The " witchcraft " (Knuth [51, 2 W. BRUNS AND A. CONCA p. 60]) of the KRS saves one from tracing the Buchberger algorithm through tedious inductions. Later on the method was extended by Herzog and Trung [44] to the so-called 1-cogenerated ideals, ladder determinantal ideals and ideals of pfaffians. They follow the important principle to derive properties of K[X ]/I t from the analogous properties of K[X ]/ in(I t): the two rings appear as the generic and special fiber of a flat 1-parameter deformation. (By in(I t) we denote the ideal of initial forms with respect to a suitable term order.) The ring K[X ]/ in(I t) is the Stanley-Reisner ring of a shellable simplicial complex and amenable to combinatorial methods …